We present a study of a class of closed Michaelis–Menten network models, which includes models of two categories of biochemical networks previously studied in the literature namely, processive and mixed mechanism phosphorylation futile cycle networks. The main focus of our study is on the uniqueness and stability of equilibrium points of this class of models. Firstly, we demonstrate that the total species concentration is a conserved quantity in models of this class. Next, we prove the existence of a unique positive equilibrium point in the set of points that correspond to a given total species concentration, using the intermediate value property of continuous functions. Finally, we demonstrate the asymptotic stability of this equilibrium point with respect to all initial conditions in the positive orthant that correspond to the same total species concentration as the equilibrium point, by constructing an appropriate Lyapunov function.

中文翻译：

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一类 Michaelis-Menten 网络的稳定性


我们提出了一类封闭式 Michaelis-Menten 网络模型的研究，其中包括先前在文献中研究的两类生化网络的模型，即持续和混合机制磷酸化无效循环网络。我们研究的主要焦点是此类模型平衡点的唯一性和稳定性。首先，我们证明总物种浓度在此类模型中是守恒量。接下来，我们使用连续函数的中间值性质证明在对应于给定总物质浓度的点集中存在唯一的正平衡点。最后，我们通过构建适当的李雅普诺夫函数，证明了该平衡点相对于与平衡点相同的总物质浓度对应的正正交中的所有初始条件的渐近稳定性。